91Ó°ÊÓ

Partial derivatives help us understand how a function changes as we tweak one of its variables while keeping others constant. In the function \(f(x, y) = x \sin y\), we can explore how each variable \(x\) and \(y\) individually influences \(f\).

For the partial derivative with respect to \(x\), denoted as \(\frac{\partial f}{\partial x}\), we consider \(y\) a constant. This leads to \(\sin(y)\), highlighting that \(f\) changes with \(x\) directly proportional to \(\sin(y)\).

On the other hand, the partial derivative with respect to \(y\), \(\frac{\partial f}{\partial y}\), treats \(x\) as constant, resulting in \(x \cos(y)\). This shows how the change in \(f\) due to \(y\) depends on both \(x\) and \(\cos(y)\). Partial derivatives provide a crucial tool for evaluating the function's slope in specific directions.

Differential approximation offers a practical way to estimate small changes in a function's output. This concept utilizes the total differential, denoted as \(dz\), to make such approximations feasible.

The formula for the total differential is given by:

• \(dz = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy\)

The expression combines partial derivatives and small changes in each variable \(dx, dy\).

In practice, when we calculate \(dz\), it serves as an approximation for the actual change \(\Delta z\). Although not exact, this method is particularly helpful for scenarios involving minuscule variations. Hence, by substituting \(dx = 0.05\) and \(dy = 0.1\) into \(dz\), we can estimate how \(f\) reacts to slight shifts in \(x\) and \(y\).

Trigonometric functions like sine and cosine are pivotal in modeling periodic phenomena. In the context of the function \(f(x, y) = x \sin y\), \(\sin(y)\) embodies one of these essential functions.

Key features of sine include:

• It ranges between -1 and 1.
• \(\sin(y)\) completes a full cycle over \(2\pi\) radians.
• The function oscillates smoothly, producing a wave-like graph.

Sine's behavior at specific values, such as \(y = 2\) or \(y = 2.1\), affects the overall value of \(f(x, y)\). If \(\sin(y)\) increases, so does \(f(x, y)\), demonstrating its direct influence, especially relevant in the exercise at hand.

When evaluating a function, the goal is to determine its output for given inputs. In the example \(f(x, y) = x \sin y\), we plug specific values of \(x\) and \(y\) into the equation.

Consider the scenarios:

• For \(f(1,2)\), substitute \(x = 1\) and \(y = 2\), yielding \(1 \times \sin(2)\).
• For \(f(1.05, 2.1)\), use \(x = 1.05\) and \(y = 2.1\), resulting in \(1.05 \times \sin(2.1)\).

Calculation of these outputs allows us to observe \(\Delta z\) by comparing the difference \(f(1.05, 2.1) - f(1, 2)\). Function evaluation is a fundamental mathematical task, vital for analyzing how changes in variables reflect on the function's output.